Find an nth-degree polynominal function with real coefficients satisfying the given conditions. L

Question

N=3; -2 and 7+5i are zeros; f(2)=200

f(x)=

Answer 1

n=3 means that the polynomial is a third degree polynomial, so the largest exponential term is x³. A degree three polynomial has three roots. You are given two: -2 and 7 + 5i. With real coefficients, if 7 + 5i is a root, so is its conjugate, 7 - 5i. Now you have all of the roots, so use the factored form"

f(x) = a(x-q)(x-r)x-s)

where q = -2, r = 7+5i, s = 7-5i

f(x) = a(x-(-2))(x-(7+5i))(x-(7-5i))

f(x) = a(x+2)(x-7-5i)(x-7+5i)

To find the value of teh constanta, plug in the given point f(2) = 200 and solve for a::

f(2) = a(2+2)(2-7-5i)(2-7+5i)

200 = a(4)(-5-5i)(-5+5i)

200 = a(200)

1 = a

The final equation in factored form is:

f(x) = (x+2)(x-7+5i)(x-7-5i)

Multiply (FOIL) it out if you want it in standard form.